Problems

Show that, if \(n > 1\), \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\) is not an integer. [Hint. Take \(m\), the largest integer such that \(2^m \leq n\) and split the sum as $$1 + \frac{1}{2} + \ldots + \frac{1}{2^m} + \frac{1}{3} + \frac{1}{5} + \frac{1}{6} + \ldots]$$

Solve the following equations completely: \begin{align} 2x + 5y + z &= a,\\ x + 3y + az &= 1,\\ 3x + 8y + bz &= c. \end{align} In particular, for what values of \(a\), \(b\), \(c\) have these equations

1. [(i)] no solutions,
2. [(ii)] more than one solution?

A monomial of degree \(n\) in the \(m\) variables \(x_1, x_2, \ldots, x_m\) is defined to be an expression of the form $$x_1^{t_1} \ldots x_m^{t_m}$$ where each of \(t_1, \ldots, t_m\) is a non-negative integer and \(t_1 + \ldots + t_m = n\). Find the number of monomials of degree \(n\) in \(m\) variables, and show that the number of monomials of degree \(\leq n\) in \(m\) variables is $$\frac{(m+n)!}{m! \, n!}.$$

The equation $$ax^4 + bx^3 + cx^2 + dx + e = 0,$$ where \(a\) and \(e\) are not zero, has roots \(\alpha, \beta, \gamma, \delta\). Show how it is possible to obtain \(\alpha^n + \beta^n + \gamma^n + \delta^n\) in terms of the coefficients \(a, b, c, d, e\) for all values of \(n\), where \(n\) is a positive or negative integer. Obtain the equations whose roots are

1. [(i)] \(\alpha^2, \beta^2, \gamma^2, \delta^2\);
2. [(ii)] \(\alpha - 3, \beta - 3, \gamma - 3, \delta - 3\).

Let \(f(x)\) and \(g(x)\) be polynomials of degree \(m\), \(n\) respectively. Show that $$f(x) = q(x)g(x) + r(x)$$ where \(q(x)\) and \(r(x)\) are polynomials, and \(r(x)\) either is zero or has degree less than \(n\). Show also that \(q(x)\) and \(r(x)\) are determined uniquely by \(f(x)\) and \(g(x)\). Hence or otherwise show that a polynomial of degree \(n\) has at most \(n\) roots.

Three complex numbers \(z_1, z_2, z_3\) are represented in the complex plane by the vertices of a triangle \(A_1A_2A_3\). What is the locus of points representing the complex numbers \(z_1 + it(z_2 - z_3)\), where \(t\) is a real parameter? Prove that the orthocentre of the triangle \(A_1A_2A_3\) represents the complex number \(z\), where $$z = \frac{\bar{z_1}(z_2 - z_3)(z_2 + z_3 - z_1) + \bar{z_2}(z_3 - z_1)(z_3 + z_1 - z_2) + \bar{z_3}(z_1 - z_2)(z_1 + z_2 - z_3)}{\bar{z_1}(z_2 - z_3) + \bar{z_2}(z_3 - z_1) + \bar{z_3}(z_1 - z_2)}$$ and the bar indicates complex conjugate.

A variable chord \(QR\) of a parabola subtends a right angle at a fixed point \(P\) of the parabola. Show that \(QR\) passes through a fixed point \(F\) on the normal at \(P\). Find the locus of \(F\) as \(P\) varies on the curve.

The end \(A\) of a line segment \(AB\) of length \(2a\) lies on the circle \(x^2 + y^2 = a^2\), and \(B\) lies on the line \(y = 0\). Show that the locus of the mid-point \(P\) of \(AB\) is the curve $$(x^2 + y^2)(x^2 + 9y^2) = 4a^2x^2.$$ Sketch this curve, indicating the relation between the position of \(B\) on the line \(y = 0\) and the position of \(P\) on the curve.

A circle touches the rectangular hyperbola \(x^2 - y^2 = a^2\) in two real points. Show that the circle belongs to one of the two families

1. [(i)] \(x^2 + (y - 2h)^2 = a^2 + 2h^2\),
2. [(ii)] \((x - 2c)^2 + y^2 = 2c^2 - a^2\) \quad \((c^2 > a^2)\).

Prove that if two circles of the second family touch, their radii \(r_1\), \(r_2\) satisfy $$r_1^2 - 6r_1r_2 + r_2^2 - 8a^2 = 0.$$

A regular tetrahedron, with edges of length \(a\), is inscribed in a sphere of radius \(R\). Find the value of the ratio \(a/R\).